One easy solution would be to build a 1-foot by 1-foot box and fill it with a DieHard battery from Sears! Or fill it with as many 9-volt Duracells as you can and wire them together. But that probably violates the spirit of the contest.
How Batteries Work contains three different types of "safe" battery chemistry for you to experiment with. The key to high power is surface area and fresh chemicals, so be sure to use big, thin sheets of metal and pour in the liquids at the competition -- not beforehand.
Here are some interesting links:
- How Batteries Work
- How do the battery testers on battery packages work?
- What is the difference between a normal lead-acid car battery and a "deep cycle" battery?